Cohen and Glashow argued that very special conformal field theories of a particular kind [i.e., with HOM(2) or SIM(2) invariance] cannot be constructed within the framework of local field theories. We, however, show examples of local construction by using nonlinear realization. We further construct linear realization from the topological twist at the cost of unitarity. To demonstrate the ubiquity of our idea, we also present corresponding holographic models.